By the structure of parabolic subgroups [1]}, the minimal parabolic subgroup \(\le \)  is equal to the semi-direct product \(_()\ltimes {u}{}\) , where \(\)  is a maximal \(k\) -split torus, and \(_()\)  is the centralizer of \(\)  in \(\) .
By [2]}, the unipotent group \({u}{}\)  is defined and split over \(k\)  and there exists a \(\) -equivariant \(k\) -isomorphism \(\Phi \colon {u}{}\rightarrow ({u}{}) \) .
Hence pulling the \(k\) -linear structure from the Lie-algebra action to the center of the unipotent radical, we get that \(Z=Z({u}{P})\)  is isomorphic to a finite-dimensional \(k\) -vector space admitting a \(T\)  linear action.
Since the torus is \(k\) -split, the linear space \(Z\)  admits a weight space decomposition with respect to non-trivial characters \(\chi _i\colon T \rightarrow k^*\) .
This means \(Z=\bigoplus _{i}V_i\)  such that for any \(v_i\in V_i\)  and \(t\in T\) , \(t.v_i=\chi _i(t)\cdot v_i\) .
Following [1]}, for any \(i\) , there exists a co-character \(\lambda _i\colon k^*\rightarrow T\)  such that the composition \(\psi _{m_i}= \chi _i\circ \lambda _i\)  is the power map
\(\psi _{m_i}\colon k^*\rightarrow k^*, \ \ \psi _{m_i}\colon x\mapsto x^{m_i}\) 
Writing \(m_i=p^{N_i}\cdot d_i\)  with \(N_i\)  equal to the \(p\) -adic valuation of \(m_i\) , we define the local field \(k_i=k^{p^{N_i}}\) . By [4]} the map
\(\psi _{d_i}\colon k^*\rightarrow k^*,\ \ \psi _{d_i}\colon x\mapsto x^{d_i}\) 
is a local homeomorphism at \(1_k\) . Moreover, the Fourbenuius map
\(\psi _{p^{N_i}}\colon k^*\rightarrow k_i^*,\ \ \psi _{p^{N_i}}\colon x\mapsto x^{p^{N_i}}\) 
is a homeomorphism from \(k^*\)  onto \(k_i^*\) .
We will show relative minimality of \({u}{P}\)  in \(A=T\ltimes {u}{P}\le P\)  and thus obtain relative minimality of \( {u}{P}\)  in \(P\) .
Let \(\)  be a group topology on \(A\)  weaker than the analytic topology.
We claim that for any \(i\) , the subgroup \(V_i\)  is minimal in \(A\) .
Following Remark REF , furnished with the analytic topology, the group \(T\)  acts continuously on \(V_i\)  endowed with the \(\) -subspace topology.
We get that the action of \(k^*\)  on \(V_i\)  defined by \(x.v=x^{m_i}v\)  is continuous, since it is the composition of the co-character \(\lambda _i\)  with the \(T\)  action via the character \(\chi _i\) .
Since \(k\)  is a finite filed extension of \(k_i\) , \(V_i\)  is a finite-dimensional \(k_i\) -vector space. Denote by \(m\colon k_i\times V_i \rightarrow V_i\)  the scalar multiplication. By the following commutative diagram
\(\begin{tikzcd}k^*\times V_i ["\pi "] ["\psi _{m_i}",swap]& \\k_i^*\times V_i ["m"] & V_i\end{tikzcd}\) 
there exists an open subgroup \(1\in W\le k^*\)  such that \(\psi _{m_i}(W)\le {k_i}^*\)  is an analytic subgroup acting continuously by multiplication on \(V_i\) .
Thus by Lemma , the topology on \(V_i\)  is equal to the analytic \(k_i\) -point topology which is equivalent to the analytic \(k\) -point topology.
We got that for any \(i\)  the subgroup \(V_i\)  is minimal in \(A\) .
Following Corollary REF , the subgroup \(Z=\oplus _i V_i\)  is minimal in \(A\) .
Therefore, the \(\) -subspace topology on \(Z\)  is equal to the analytic topology. Thus, by Lemma REF  the \(\) -subspace topology on \({u}{P}\)  is equal to the analytic topology.
